Question: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{-5x^2 + 10x + 120}{-8x^2 + 64x - 96}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {-5(x^2 - 2x - 24)} {-8(x^2 - 8x + 12)} $ $ a = \dfrac{5}{8} \cdot \dfrac{x^2 - 2x - 24}{x^2 - 8x + 12} $ Next factor the numerator and denominator. $ a = \dfrac{5}{8} \cdot \dfrac{(x - 6)(x + 4)}{(x - 6)(x - 2)}$ Assuming $x \neq 6$ , we can cancel the $x - 6$ $ a = \dfrac{5}{8} \cdot \dfrac{x + 4}{x - 2}$ Therefore: $ a = \dfrac{ 5(x + 4)}{ 8(x - 2)}$, $x \neq 6$